A Fields transforming in the adjoint representation?

[Juanchotutata]

Hi!
I'm doing my master thesis in AdS/CFT and I've read several times that "Fields transforms in the adjoint representation" or "Fields transforms in the fundamental representation". I've had courses in Advanced mathematics (where I studied Group theory) and QFTs, but I don't understand (or maybe I don't remember) why this is relevant. For example, last time I was reading about large N limit in vector theories and again the author started to talk about adding matter fields in the adjoint representation. I would be most grateful if someone can solve me this doubt.

Thanks for your time and attention! :)

 

[Orodruin]

It is a bit unclear to me exactly which part is bothering you. Can you try to be a bit more specific? Is it the definition of a field transforming according to some representation, the meaning of fundamental/adjoint representation, or something else?

 

[Juanchotutata]

Thank you for your answer Orodruin!. I'm sorry if I wasnt clear. I'm physicist so maybe is the meaning of adjoint and fundamental representation what is not clear for me...
I don't understand why the matter fields are in the adjoint representation or in the fundamental representation... and what happens with that.

Apologize me if my english is not the best.
Greetings!

 

[Orodruin]

First of all, do you have a basic understanding of what it means for a field to transform according to a particular representation of a group? If not, we must start there.

If you do, are you familiar with what the fundamental/adjoint representations are?

 

[Juanchotutata]

Mmm I know that a representation is an application which match the elements of a group with matrices. But I'm not sure if I know what you're saying about the transformation according to a particular representation.

 

[Orodruin]

Ok, so let us start from what you know, which is correct. A representation is a map from a group to a set of matrices such that the group structure is preserved, i.e., if the representation is called $\rho$, then $\rho(ab) = \rho(a) \rho(b)$, where $a$ and $b$ are group elements. The representation is $n$-dimensional if the matrices it maps to are $n\times n$ matrices. These matrices naturally act on an $n$-dimensional vector space, which is the vector space in which a matter field that transforms according to that representation lives and making a group transformation $a$ means that a matter field in that representation transforms according to $\psi \to \rho(a)\psi$.

The most common (and simplest) example is that of a $U(1)$ group, where an element can be represented by a complex number of modulus one, $e^{i\alpha}$. The corresponding representation is one-dimensional and a field in this representation would be simply a vector with one element, transforming according to $\psi \to e^{i\alpha}\psi$.

We then come to the fundamental and adjoint representations. For a matrix group, such as $SU(n)$, the fundamental representation is just $\rho(a) = a$, i.e., you represent a matrix by the matrix itself. Thus, for $SU(n)$, the fundamental representation is $n$-dimensional and a field transforming under the fundamental representation is a column vector with $n$ entries transforming according to $\psi \to \rho(a) \psi = a\psi$.

The adjoint representation is slightly more involved and we instead let the field be an element of the Lie algebra of the group with the representation given by the group's natural action on the Lie algebra.

 

[Juanchotutata]

Thank you for your elaborated answer Orodruin.

I think that I've fully understood how the field transforms according to one representation or another but I've a question from the physicist view. Are there any physical difference between a field transforming in the adjoint representation or in the fundamental representation?. Or is this just a mathematical issue?.

I think that I've read that for calculations in AdS/CFT it's easier when we consider that the matter fields transform in the adjoint representation, but I'm not sure if this is true.

Thank you again for your time. I'm new here and I don't know how I can possitively vote you.

 

[Orodruin]

Are there any physical difference between a field transforming in the adjoint representation or in the fundamental representation?. Or is this just a mathematical issue?.

Yes, there is a difference. The fundamental and adjoint representations are not the same and generally have different dimension. This leads to different numbers of degrees of freedom in your fields.